Nonlinear Amplitude Research Articles

Analytical modeling of the linear and nonlinear dynamic characteristics of the non-uniform axially functionally graded cylindrical beam based on the strain gradient theory

In this research, the nonlinear vibration behavior of axially functionally graded (AFG) nano-scale truncated conical tubes (with non-uniform cross-section), including the porosity, is presented for the first time based on the Euler-Bernoulli beam theory and the von-Kármán’s nonlinear strain-displacement and nonlinear nonlocal boundary conditions. The effect of being at the tapered nanosize model is supposedly based on the nonlocal strain gradient theory using the Hamilton principle to extract the equation of motions and nonlinear time-dependent nonlocal boundary conditions. The material properties and thickness of nano-tube are varying along the length of tubes. The generalized differential quadrature method (GDQM) is utilized with the iteration method to solve the complicated nonlinear equations. The numerical results in tabular and graphical styles present for various nonlinear amplitudes, the AFG power indexes, the rate of the cross-section change, the porosity parameter, and the nonlocal gradient strain parameter for both simply supported, fully clamped, and simply supported-clamped boundary conditions in detail.

Determining the source of phase noise: Response of a driven Duffing oscillator to low-frequency damping and resonance frequency fluctuations

We present an analytical calculation of the response of a driven Duffing oscillator to low-frequency fluctuations in the resonance frequency and damping. We find that fluctuations in these parameters manifest themselves distinctively, allowing them to be distinguished. In the strongly nonlinear regime, amplitude and phase noise due to resonance frequency fluctuations and amplitude noise due to damping fluctuations are strongly attenuated, while the transduction of damping fluctuations into phase noise remains of order 1. We show that this can be seen by comparing the relative strengths of the amplitude fluctuations to the fluctuations in the quadrature components, and suggest that this provides a means to determine the source of low-frequency noise in a driven Duffing oscillator.

Non-linear free and forced vibration of bi-directional functionally graded truncated conical tube based on the nonlocal gradient strain theory

This paper deals with the free and forced, linear and nonlinear vibration of functionally graded (along with thickness) nanotube, considering the non-uniform cross-section that made a two-dimensional functionally graded (2D-FG) structure. The bi-dimensional nanostructure-dependent governing equation is modeled based on the classic beam theory (Euler-Bernoulli), and they are derived by Hamilton’s principle based on nonlocal gradient strain theory considering the von-Kármán nonlinear strain and the external harmonic load. To solve the general equations and calculate the results, the generalized differential quadrature method (GDQM) was used coupled with the numerical iteration method for the nonlinear response. The results are discussed for the different simply-supported and clamped boundary conditions. The 2D-FG nanotube’s behavior is analyzed for different nonlinear amplitudes, nonlocal strain gradient parameters and nonlocal parameters, different rates of cross-sections, and different material distributions.

Magneto-electro-elastic modelling and nonlinear vibration analysis of bi-directional functionally graded beams

In the paper, a novel magneto-electro-elastic model of bi-directional (2D) functionally graded materials (FGMs) beams is developed for investigating the nonlinear dynamics. It is shown that the asymmetric modes induced by the 2D FGMs may significantly affect the nonlinear dynamic responses, which is tremendously different from previous studies. Taking into account the geometric nonlinearity, the nonlinear equation of motion and associated boundary conditions for the beams are derived according to the Hamilton’s principle. The natural frequencies and numerical modes of the beams are calculated by the generalized differential quadrature method. The frequency responses of the nonlinear forced vibration are constructed based on the Galerkin technique incorporating with the incremental harmonic balance approach. The influences of the material distributions, length–thickness ratio, electric voltage, magnetic potential as well as boundary condition on the nonlinear resonant frequency and response amplitude are discussed in details. It is notable that increasing in the axial and thickness FG indexes, negative electric potential and positive magnetic potential can lead to decline the nonlinear resonance frequency and amplitude peak, which is usually applied to accurately design the multi-ferroic composite structures. Furthermore, the nonlinear characteristics of motion can be regulated by tuning/tailoring the 2D FG materials.

Nonlinear forced vibration analysis of functionally graded non-uniform cylindrical microbeams applying the semi-analytical solution

The linear and nonlinear forced vibration response of axially functionally graded (AFG) cylindrical truncated conical and imperfect microbeam subjected to the dynamic harmonically load carried out in the presented research. Based on a couple of modified couple stress theory, the Euler-Bernoulli beam theory and von-Kármán theory, the linear and nonlinear governing equations and related boundary conditions for dynamic response of micro-size tubes are derived employing the Hamilton principle. We considered the uniform and nonuniform functions for the cross-section, in which the convex, linear and exponential functions are the nonuniform sections, and the porosity is regarded as an imperfection. The generalized differential quadrature method (GDQM) is used to prepare the initial conditions for homotopy perturbation (HP) techniques as the semi-analytical approach to calculate the linear and nonlinear results of dynamic responses. The obtained linear and nonlinear results of the free and forced vibration response show the negative and positive effects of some parameters such as the porosity parameter, the nonlinear amplitude, the small-scale parameter, AFG parameter, and different cross-section impact on the dynamic deflection and natural frequency of micro-scale tube and beams with both clamped and simply-supported boundary conditions.

Dark states and vortex solitary wave with finite energy of Maxwell-Dirac nonlinear equations

Starting from the Maxwell’s equations for media with no-stationary linear and nonlinear polarization, we obtain a set of nonlinear Maxwell amplitude equations in the approximation of the first order of dispersion. After a special kind of complex presentation, the set of amplitude equations is written as a set of nonlinear Dirac equations. For broad-band pulses solitary solutions with half-integer spin and finite energy are found. The solutions correspond to an electromagnetic wave with circular Poynting vector and zero divergence. These invisible for the detectors waves are called dark states and the localized energy is determinates as electromagnetic mass.

Influence of cross-section on the linear and nonlinear buckling analysis of imperfect functionally graded micro-tubes

This article deals with the linear and nonlinear buckling behavior of porosity-dependent functionally graded (FG) non-uniform microtubes. The modified couple stress theory is determined to consider small-scale effects on the micro-size tube which is modeled, based on the Euler–Bernoulli beam theory in conjunction with the employment of the conservation energy principle to derive the nonlinear governing equations regarding the nonlinear Von-Kármán strains. The main contribution of this article is the impact of cross-sections on the static behavior of cylindrical beams with four applicable cross-sections, involving the exponential, linear, and convex in comparison of uniform section for the radial distribution of ceramic and metal material based on the power-law function. Also, the pinned, clamped, and combination of these boundary conditions are defined as the boundary supports, and the generalized quadrature method (GDQM) is utilized as the numerical solving technique along with the iteration method to solve the nonlinear equations. Finally, the obtained results extracted to investigate the impact of various parameters, involving the nonlinear amplitude, power FG index, porosity parameter, small-scale parameter, and different cross-sections on the buckling treatment of cylindrical beam.

Label-free detection of brain tumors in a 9L gliosarcoma rat model using stimulated Raman scattering-spectroscopic optical coherence tomography.

.Significance: In neurosurgery, it is essential to differentiate between tumor and healthy brain regions to maximize tumor resection while minimizing damage to vital healthy brain tissue. However, conventional intraoperative imaging tools used to guide neurosurgery are often unable to distinguish tumor margins, particularly in infiltrative tumor regions and low-grade gliomas.Aim: The aim of this work is to assess the feasibility of a label-free molecular imaging tool called stimulated Raman scattering-spectroscopic optical coherence tomography (SRS-SOCT) to differentiate between healthy brain tissue and tumor based on (1) structural biomarkers derived from the decay rate of signals as a function of depth and (2) molecular biomarkers based on relative differences in lipid and protein composition extracted from the SRS signals.Approach: SRS-SOCT combines the molecular sensitivity of SRS (based on vibrational spectroscopy) with the spatial and spectral multiplexing capabilities of SOCT to enable fast, spatially and spectrally resolved molecular imaging. SRS-SOCT is applied to image a 9L gliosarcoma rat tumor model, a well-characterized model that recapitulates human high-grade gliomas, including high proliferative capability, high vascularization, and infiltration at the margin. Structural and biochemical signatures acquired from SRS-SOCT are extracted to identify healthy and tumor tissues.Results: Data show that SRS-SOCT provides light-scattering-based signatures that correlate with the presence of tumors, similar to conventional OCT. Further, nonlinear phase changes from the SRS interaction, as measured with SRS-SOCT, provide an additional measure to clearly separate tumor tissue from healthy brain regions. We also show that the nonlinear phase signals in SRS-SOCT provide a signal-to-noise advantage over the nonlinear amplitude signals for identifying tumors.Conclusions: SRS-SOCT can distinguish both spatial and spectral features that identify tumor regions in the 9L gliosarcoma rat model. This tool provides fast, label-free, nondestructive, and spatially resolved molecular information that, with future development, can potentially assist in identifying tumor margins in neurosurgery.

Exploring the gelation of aqueous cellulose nanocrystals (CNCs)-hydroxyethyl cellulose (HEC) mixtures

We investigated the gelation and microstructure of cellulose nanocrystals (CNCs) in nonionic hydroxyethyl cellulose (HEC) solutions. Cellulose nanocrystals (CNCs) with a particle length of 90 nm and width of 8 nm currently produced by acid hydrolysis of wood pulp were used in this study. The microstructures of CNCs/polymer suspensions were investigated by performing linear small amplitude oscillatory shear (SAOS) and nonlinear large amplitude oscillatory shear (LAOS), in addition to constructing CNCs phase diagrams and measuring steady-state shear viscosities. Significant viscosity increases at low shear rates coupled with high shear thinning behaviors were observed in CNCs in HEC solutions above the overlapping concentration of HEC. The physical strength of CNCs/HEC solution gels increased with the increase in CNCs concentration and resembled the weakly crosslinked gels according to the scaling of linear dynamic mechanical experiments. According to LAOS analysis, CNCs/HEC mixtures showed type III behavior with intercycle stress softening, while the samples showed stress stiffening in single cycles.Graphical abstract

Implementing a Magnonic Reservoir Computer Model Based on Time-Delay Multiplexing

Recently we demonstrated experimentally that microwave oscillators based on the time delay feedback provided by traveling spin waves could operate as reservoir computers. In the present paper, we extend this concept by adding the feature of time multiplexing made available by the large propagation times/distances of traveling spin waves. The system utilizes the nonlinear behavior of propagating magnetostatic surface spin waves in a yttrium-iron garnet thin film and the time delay inherent in the active ring configuration to process time dependent data streams. Higher reservoir dimensionality is obtained through the time-multiplexing method, emulating "virtual" neurons as temporally separated spin-wave pulses circulating in the active ring below the auto-oscillation threshold. To demonstrate the efficacy of the concept, the active ring reservoir computer is evaluated on the short-term memory and parity check benchmark tasks, and the physical system parameters are tuned to optimize performance. By incorporating a reference line to mix the input signal directly onto the active ring output, both the amplitude and phase nonlinearity of the spin waves can be exploited, resulting in significant improvement on the nonlinear parity check task. We also find that the fading memory capacity of the system can be easily tuned by controlling the active ring gain. Finally, we show that the addition of a second spin-wave delay line configured to transmit backward volume spin waves can partly compensate dispersive pulse broadening and enhance the fading memory capacity of the active ring.

Improving damping capabilities of composites structures by electroactive films containing piezoelectric and conductive fillers

In this paper, a passive vibration damping concept based on multifunctional materials was studied for thermoplastic composite structures. The synergy between piezoelectric and conductive particles brings a new contribution of energy dissipation based on the local transduction-dissipation phenomenon. While piezoelectric fillers ensure the conversion of mechanical energy into electrical energy (transduction), conductive particles locally dissipate the electric charges created avoiding saturation in the vicinity of piezoelectric particles. Here, the concept has been studied at material and structure scales for laboratory and preindustrial samples in order to bring solid proof of the damping concept. For this purpose, piezoelectric and electrically conductive particles were dispersed into engineering thermoplastics polyamide 12 and poly ether ketone ketone. Damping films were obtained by hot press and embedded in a composite sandwich beam and carbon fiber reinforced polymer (CFRP)-aluminum panels. Dynamic mechanical analysis and vibration tests were performed on bulk nanocomposite samples and in composite sandwich beams. The study of hysteresis loops and frequency response function showed strong nonlinear effects and vibration amplitude decrease up to 50%. Tests on CFRP-aluminum panels highlighted the structural damping increase demonstrating the potential capacity of this multifunctional material for energy dissipation in typical aerospace structures.

Interaction of equivalence ratio fluctuations and flow fluctuations in acoustically forced swirl flames

The generation and turbulent transport of temporal equivalence ratio fluctuations in a swirl combustor are experimentally investigated and compared to a one-dimensional transport model. These fluctuations are generated by acoustic perturbations at the fuel injector and play a crucial role in the feedback loop leading to thermoacoustic instabilities. The focus of this investigation lies on the interplay between fuel fluctuations and coherent vortical structures that are both affected by the acoustic forcing. To this end, optical diagnostics are applied inside the mixing duct and in the combustion chamber, housing a turbulent swirl flame. The flame was acoustically perturbed to obtain phase-averaged spatially resolved flow and equivalence ratio fluctuations, which allow the determination of flux-based local and global mixing transfer functions. Measurements show that the mode-conversion model that predicts the generation of equivalence ratio fluctuations at the injector holds for linear acoustic forcing amplitudes, but it fails for non-linear amplitudes. The global (radially integrated) transport of fuel fluctuations from the injector to the flame is reasonably well approximated by a one-dimensional transport model with an effective diffusivity that accounts for turbulent diffusion and dispersion. This approach however, fails to recover critical details of the mixing transfer function, which is caused by non-local interaction of flow and fuel fluctuations. This effect becomes even more pronounced for non-linear forcing amplitudes where strong coherent fluctuations induce a non-trivial frequency dependence of the mixing process. The mechanisms resolved in this study suggest that non-local interference of fuel fluctuations and coherent flow fluctuations is significant for the transport of global equivalence ratio fluctuations at linear acoustic amplitudes and crucial for non-linear amplitudes. To improve future predictions and facilitate a satisfactory modelling, a non-local, two-dimensional approach is necessary.

Global gyrokinetic simulation of neoclassical ambipolar electric field and its effects on microturbulence in W7-X stellarator

Global neoclassical simulations of a model equilibrium of the W7-X stellarator find an ambipolar electric field with either an ion root or an electron root solution depending on the temperature ratio between electrons and ions. The ambipolar electric field is then used as an equilibrium field in the turbulence simulations of ion temperature gradient (ITG) instability. The shear of the ambipolar electric field has modest effects on the ITG linear instability, nonlinear saturation amplitude, and turbulent transport in the ion root case. However, in the electron root case, the ambipolar electric field significantly reduces the linear ITG growth rate, turbulence intensity, and radial correlation length, resulting in an ion heat conductivity comparable to the neoclassical transport level in the strong shear region.

Simple Behavioral Model of Baseband Pulse Devices in the Form of a Second-Order Nonlinear Recursive Filter

A minimalistic nonlinear dynamic model, which still reflects the main aspects of the devices response to baseband pulse signals, is considered. The model is built as “black-box” and does not require knowing the structure and physical properties of the device. Since the model should reflect the exponential-type transient response, it is based on a nonlinear recursive filter. In the recursive loops the integrators are placed, because the main effect in the response of the simulated devices is the smoothing of the signal edge. The model reflects the nonlinear amplitude characteristic of the device (at the flat top of pulse), the dependence of the transient process duration on the input signal, and the nonlinear properties of the overshoot at the flat top of the pulse. We consider the realization of this model in CAD system and the methods for extracting the model parameters. The simulation error with the proposed model is 2-10% depending on the similarity of the test signal and signal at which the parameters were extracted.

The homotopy perturbation method for fractional differential equations: part 2, two-scale transform

PurposeThe purpose of this paper is to find an approximate solution of a fractional differential equation. The fractional Newell–Whitehead–Segel equation (FNWSE) is used to elucidate the solution process, which is one of the nonlinear amplitude equation, and it enhances a significant role in the modeling of various physical phenomena arising in fluid mechanics, solid-state physics, optics, plasma physics, dispersion and convection systems.Design/methodology/approachIn Part 1, the authors adopted Mohand transform to find the analytical solution of FNWSE. In this part, the authors apply the fractional complex transform (the two-scale transform) to convert the problem into its differential partner, and then they introduce the homotopy perturbation method (HPM) to bring down the nonlinear terms for the approximate solution.FindingsThe HPM makes numerical simulation for the fractional differential equations easy, and the two-scale transform is a strong tool for fractal models.Originality/valueThe HPM with the two-scale transform sheds a bright light on numerical approach to fractional calculus.

Research on nonlinear vibration control of laminated cylindrical shells with discontinuous piezoelectric layer

A model of laminated cylindrical shells with discontinuous piezoelectric layer is proposed. Based on the first-order shear nonlinear shell theory, the nonlinear vibration control of the piezoelectric laminated cylindrical shell model with point-supported elastic boundary condition is analyzed. In this model, a series of artificial springs are introduced to simulate the arbitrary boundary conditions. And the elastic-electrically coupled differential equations of piezoelectric laminated cylindrical shells are obtained by the Chebyshev polynomials and Lagrange equations and decoupled by using the negative velocity feedback adjustment. Then, the frequency–amplitude responses of the piezoelectric laminated cylindrical shells are obtained by the incremental harmonic balance method. Finally, the influence of the constant gain, size, and position of the piezoelectric layer on the nonlinear amplitude–frequency response is investigated. The results show that the constant gain and the position and size of the piezoelectric layer have a significant influence on the amplitude of the nonlinear amplitude–frequency and time–frequency response.

A Mobile Mathieu Oscillator Model for Vibrational Locomotion of a Bristlebot

Abstract Terrestrial locomotion that is produced by creating and exploiting frictional anisotropy is common amongst animals such as snakes, gastropods, and limbless lizards. In this paper we present a model of a bristlebot that locomotes by generating frictional anisotropy due to the oscillatory motion of an internal mass and show that this is equivalent to a stick–slip Mathieu oscillator. Such vibrational robots have been available as toys and theoretical curiosities and have seen some applications such as the well-known kilobot and in pipe line inspection, but much remains unknown about this type of terrestrial locomotion. In this paper, motivated by a toy model of a bristlebot made from a toothbrush, we derive a theoretical model for its dynamics and show that its dynamics can be classified into four modes of motion: purely stick (no locomotion), slip, stick–slip, and hopping. In the stick mode, the dynamics of the system are those of a nonlinear Mathieu oscillator and large amplitude resonance oscillations lead to the slip mode of motion. The mode of motion depends on the amplitude and frequency of the periodic forcing. We compute a phase diagram that captures this behavior, which is reminiscent of the tongues of instability seen in a Mathieu oscillator. The broader result that emerges in this paper is that mobile limbless continuum or soft robots can exploit high-frequency parametric oscillations to generate fast and efficient terrestrial motion.

Fast and Precise Control for the Vibration Amplitude of an Ultrasonic Transducer Based on Fuzzy PID Control.

This work presents a novel constant frequency ultrasonicamplitude control (CFUAC) method based on fuzzy proportional-integral-derivative (FPID) and amplitude direct feedback. The frequency shift and amplitude nonlinearity of the piezoelectric transducer (PT) are measured to determine the optimal constant control frequency of 19.2 kHz. The FPID controller is designed to adapt to the nonlinear changes in different target amplitudes and loads. A direct PT amplitude feedback method is used to improve the signal's anti-interference ability and accuracy. The 5% settling time and steady-state error of FPID can reach 92.22 ms and [Formula: see text] at the step response under 24 [Formula: see text]. The 5% settling time and steady-state error of FPID are less than 131.44 ms and [Formula: see text] at 10 [Formula: see text] under 150 N. The results confirmthat fast and precise control of the vibration amplitude of an ultrasonic transducer can be realized by the proposed method. The new CFUAC method lays a foundation for revealing the ultrasonic welding and metal processing (UWMP) mechanism and helps to expand the application of ultrasonic vibration in the fields of precision machining and high dynamic ultrasonicmedical equipment.

Delay-Robust Nonlinear Control of Bounded-Input Telerobotic Systems With Synchronization Enhancement

This letter puts forward a novel controller for joint position tracking of bilateral teleoperation systems subjected simultaneously to time-varying communication delays and bounded actuation. Enhancing such systems' robustness to the larger time delays comes prevalently at the cost of increased settling time for position synchronization. To this end, we propose a general and refined form of nP+D controller that not only mitigates the trade-off between settling time of synchronization and magnitude of time-delay but also exhibits better transient error in position convergence. These advantages are brought along through using capped joint-velocity in the controller, which offers a blessing in disguise in our presented Lyapunov-based stability analysis and allows disposing of the limitation that was originally considered on the nonlinear function's amplitude in previous nP+D controllers. We have shown that by setting conditions on the controller parameters obtained from the analytical study, the closed-loop dynamics' asymptotic stability is ensured. The proposed controller's efficacy and outperformance are validated through numerical simulations and experimental evaluations on a bilateral teleoperation system with multi-DOF robots as the leader and follower.

Geometrically nonlinear free vibration of Euler-Bernoulli shallow arch

The purpose of this work is to investigate the geometrical non-linearity in free vibrations of the Euler-Bernoulli shallow arch with clamped ends. The nonlinear governing equilibrium equation of the shallow arch is obtained after the Euler Bernoulli theory and Won Karman geometrical nonlinearity assumptions. The initial curvature of the arch is not due to the axial displacement of the beam but is due to the geometric of the beam itself. Taking into account a harmonic motion, the kinetic and total strain energy are discretized into a series of finite functions, which are a combination of the linear modes calculated before and the coefficients of contribution. The discretized expressions are derived by applying a Hamilton principle energy and spectral analysis. A cubic nonlinear algebraic system is obtained and solved numerically using an approximation method (the so-called second formulation) is applied to resolve various nonlinear vibration problems. To illustrate the effect of the curvature on the fundamental nonlinear mode and nonlinear frequency, the corresponding backbone curves, nonlinear amplitude vibration, and curvature of the arch are presented for the first modes shapes.